We now discuss some closely related lower bounds indicating that Theorem REF  and Theorem REF  cannot be improved in a certain sense. First, [1]} shows that if \(d \le n\) , then for any distribution of the covariates \(X\)  such that the sample covariance matrix is invertible almost surely and any linear predictor \(\tilde{\omega }\) , there is a joint distribution \((X, Y)\)  with independent zero mean Gaussian noise \(\xi \)  such that the following holds:
\(\operatorname{\mathbb {E}}R(\tilde{\omega }) - R(\omega ^*_{\infty }) \ge \operatorname{\mathbb {E}}\xi ^2\left(\operatorname{\mathbb {E}}\frac{X^{\mathsf {T}}\widehat{\Sigma }_{0}^{-1}X}{n - X^{\mathsf {T}}\widehat{\Sigma }_{0}^{-1}X}\right)= \operatorname{\mathbb {E}}\left(\frac{\xi ^2X^{\mathsf {T}}\widehat{\Sigma }_{0}^{-1}X}{n - X^{\mathsf {T}}\widehat{\Sigma }_{0}^{-1}X}\right),\) 
